A conic (or conic section) is a plane curve
that can be obtained by intersecting a cone with a plane that
does not go
through the vertex of the cone. There are three possibilities,
depending on the relative position of the cone and the plane (Figure
1).

If no line of the cone is parallel to the plane, the intersection
is a closed curve, called an ellipse.

If one line of the cone
is parallel to the plane, the intersection is an open curve whose
two ends are asymptotically parallel; this is called a parabola.

Finally, there may be two lines in the cone parallel to the plane;
the curve in this case has two open pieces, and is called a hyperbola.

Historyof Conic Sections

Conic sections are among the oldest curves,
and is an old mathematics topic studied systematically and thoroughly.
The conics seem to have been discovered by Menaechmus (a Greek,
c.375-325 BC), tutor to Alexander the Great. They were conceived
in an attempt to solve the three famous problems of trisecting
the angle, duplicating the cube, and squaring the circle. The
conics were first defined as the intersection of: a right circular
cone of varying vertex angle; a plane perpendicular to an element
of the cone. (An element of a cone is any line that makes up the
cone) Depending on whether the angle is less than, equal to, or greater than
90 degrees, we get ellipse, parabola, or hyperbola respectively.
Appollonius (c. 262-190 BC) (known as The Great Geometer) consolidated
and extended previous results of conics into a monograph Conic
Sections, consisting of eight books with 487 propositions. Quote
from Morris Kline: "As an achievement it [Appollonius' Conic
Sections] is so monumental that it practically closed the subject
to later thinkers, at least from the purely geometrical standpoint."
Book VIII of Conic Sections is lost to us. Appollonius' Conic
Sections and Euclid's Elements may represent the quintessence
of Greek mathematics.

Appollonius was the first to base the theory
of all three conics on sections of one circular cone, right or
oblique. He is also the one to give the name ellipse, parabola,
and hyperbola.

In Renaissance, Kepler's law of planetary
motion, Descarte and Fermat's coordinate geometry, and the beginning
of projective geometry started by Desargues, La Hire, Pascal pushed
conics to a high level. Many later mathematicians have also made
contribution to conics, especially in the development of projective
geometry where conics are as fundamental objects as circles are in Greek
geometry. Among the contributors, we may find Newton, Dandelin,
Gergonne, Poncelet, Brianchon, Dupin, Chasles, and Steiner. Conic
sections is a rich classic topic that has spurred many developments
in the history of mathematics.